# Adjusting arrows in a commutative diagram with tikzcd

I'm trying to type the following diagram with tikzcd:

I end up with

by the following codes:

\documentclass{article}
\usepackage{tikz, tikz-cd}
\usetikzlibrary{matrix,decorations.pathmorphing,arrows}
\tikzset{commutative diagrams/.cd,arrow style=tikz,diagrams={>=stealth'}}

\begin{document}
$\begin{tikzcd} M\times N \arrow{r}{\iota} \arrow[swap]{dr}{\varphi} & M\otimes_RN \arrow{d}{\Phi} \\ & L \end{tikzcd}$
\end{document}


How can I adjust the position of the downright arrow? How can I change the arrow tip to be the one in the first picture?

The Computer Modern Rightarrow arrow tip from the arrows.meta library looks a bit like the one in your image, if you modify its width and length, i.e.

>={Computer Modern Rightarrow[length=5pt,width=3pt]}


To move the arrow, you can for example make use the start anchor and end anchor keys, which are described in the manual.

\documentclass{article}
\usepackage{tikz-cd}
\usetikzlibrary{arrows.meta} % not arrows, but arrows.meta
\tikzset{
commutative diagrams/.cd,
arrow style=tikz,
diagrams={>={Computer Modern Rightarrow[length=5pt,width=3pt]}},
}

\begin{document}
$\begin{tikzcd} M\times N \arrow{r}{\iota} \arrow[swap, start anchor={[shift={(-3pt,3pt)}]south east}, % fine tune start of arrow end anchor={[shift={(2pt,-4pt)}]north west} % fine tune end of arrow ]{dr}{\varphi} & M\otimes_RN \arrow{d}{\Phi} \\ & L \end{tikzcd}$
\end{document}